Twistings and Hopf Galois extensions

Abstract

Let k be a commutative ring, let H be a k-Hopf algebra, and let A be a right H-comodule algebra. A twisting of A is a map τ: H⊗A→A such that (A,*τ,ρA) is also an H-comodule algebra, where the product *τ is defined by a*τb=∑a0τ(a1⊗b). In this note, we observe that there is a map of pointed sets from the twistings of A to the H-measurings from AcoH to A and study the set of twistings that map to the trivial measuring. If A/AcoH is Galois and H is finitely generated projective, then the twistings that map to the trivial measuring can be described as a set of invertible twisted cocycles: φ: H⊗H→A. An equivalence relation on the set of twisted cocycles corresponds to isomorphism classes of Galois extensions. © 2000 Academic Press.